Bag‑and‑Whisker Plot: The New Way to Visualize Bivariate Data?

Imagine you want to summarise two variables at once, like height and weight of a class. The classic boxplot works for a single variable; for two, statisticians invented the bagplot, a kind of “box with a bag”. However, the bagplot has two flaws: it uses a fixed inflation factor to spot outliers, which doesn’t adapt to sample size, and it draws a “bolster” using a convex hull that can change shape unpredictably.

The new bag‑and‑whisker plot fixes both issues. First, outlier detection is framed as a multiple‑testing problem: the method automatically decides how wide the fences should be, controlling statistical error rates. Thus, with a small sample the threshold is lenient; with a large sample it becomes stricter.

Second, the visualisation drops the bolster. Instead of a convex hull, the graph directly shows the statistical fence and adds thin whiskers that illustrate data spread. The result is a more stable and easier‑to‑read picture.

Simulations and real‑world examples show that the bag‑and‑whisker is more adaptive and robust than the traditional bagplot. For anyone handling bivariate data, it’s now a recommended tool for analysis and presentation.

The bag‑and‑whisker plot originates from a formal reformulation of the outlier‑identification problem in bivariate data. Traditionally, the bagplot uses a constant inflation factor \(c\) to build the fence around the bag, defining outliers as points lying beyond \(c\) times the bivariate interquartile range. This approach ignores sample variability: with few observations, \(c\) can be overly restrictive, while with many observations it may become too permissive.

In the new approach, fence construction is treated as a multiple‑testing problem. All points are considered null hypotheses of belonging to the bivariate “core”. By applying false discovery rate (FDR) or family‑wise error rate (FWER) control procedures, the method determines an adaptive threshold \(c_{ad}\) that varies with sample size and observed distribution. Practically, Mahalanobis distances of each point from the bag centre are computed, yielding p‑values; then, for example, the Benjamini–Hochberg procedure sets a significance cutoff. This guarantees that the overall probability of mistakenly labeling a point as an outlier stays bounded.

Graphically, the bag‑and‑whisker discards the bolster—the convex hull that wraps the bag. The bolster is known for instability: tiny data changes can cause large hull shape variations, confusing interpretation. Instead, the fence is drawn as a closed curve based on Mahalanobis distance quantiles, while whiskers are line segments extending from the bag centre toward the most extreme points along the principal variance directions (eigenvectors of the covariance matrix). These whiskers provide an intuitive spread representation, akin to univariate boxplot whiskers, but extended to the bivariate case.

Simulations reveal that, compared with the classic bagplot, the new method retains a 15‑20% higher outlier detection power in small‑sample scenarios (n<50) and reduces false‑positive rates by about 10% in large samples (n>200). Real‑world applications—such as astronomical spectrometer measurements and gene‑expression datasets—confirm greater consistency in anomaly identification and a more stable visualisation, easing interpretation for researchers and analysts.

In summary, the bag‑and‑whisker plot merges rigorous statistical correction for outlier detection with a more reliable graphical representation, making it an ideal tool for exploratory analysis of bivariate data.